Part I: You Must Solve Manually Using Calculator Question
Part I You Must Solve Manually Using Calculatorpart 1 Questions To
Part I You Must Solve Manually Using Calculatorpart 1 Questions To
Paper For Above instruction
Introduction
The set of questions in Part I focuses on applying statistical and analytical techniques—namely covariance, correlation, simple linear regression, least squares regression, and logistic regression—using calculator-based manual calculations. These problems involve interpreting data, developing models, and deriving predictions or estimates pertinent to practical scenarios such as gaming strategies, computer output relationships, and manufacturing odds ratios. Mastery of these techniques enhances analytical problem-solving skills critical in data-driven decision-making contexts.
Question 1: Covariance and Correlation Between Customers and Profit for Game 2
Given the data collected by Mr. Biden for two different games, with the covariance of Game 1 known as "H". The question asks to find the covariance and correlation between customers and profit for Game 2, in terms of H.
Approach: To solve this, we need the data from tables 1 and 2, specifically the customer and profit figures for Game 2. Assuming the data points are provided, covariance and correlation are calculated using the formulas:
- Covariance: \( \text{Cov}(X, Y) = \frac{1}{n} \sum_{i=1}^n (X_i - \bar{X})(Y_i - \bar{Y}) \)
- Correlation: \( r = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y} \)
Since the data is not explicitly provided here, but the covariance of Game 1 is known as "H", and the relationship is to be expressed in terms of "H," then the covariance of Game 2 can be calculated via the relationships between the datasets, possibly involving proportionality or covariance transformation formulas. Similarly, the correlation coefficient depends on covariance and standard deviations, which relate to the variance, and thus are derivable from the given data and known covariance "H."
Question 2: Simple Linear Regression for Input and Output Data
Mr. Kumar seeks to analyze the relationship between input (independent variable) and output (dependent variable) of his computer. Using the provided data, develop a simple linear regression model:
Regression equation: \( \hat{Y} = a + bX \)
Where:
- \(b = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2}\)
- \(a = \bar{Y} - b \bar{X}\)
After calculating \(a\) and \(b\), to predict outputs for \(X = 7\) and \(X = 8\), substitute these values into the regression equation.
For the errors, the sum of squared errors (SSE) is computed as:
\( \text{SSE} = \sum_{i=1}^n (Y_i - \hat{Y}_i)^2 \)
and the total sum of squares (SST) as:
\( \text{SST} = \sum_{i=1}^n (Y_i - \bar{Y})^2 \)
Finally, coefficient of determination \( R^2 = 1 - \frac{\text{SSE}}{\text{SST}} \) evaluates the model's goodness of fit.
Question 3: Least Squares Regression and Residuals
Given data points \((x_1, y_1)\), \((x_2, y_2)\), ... , the least squares regression line is derived by solving the normal equations:
\( \hat{Y} = \beta_0 + \beta_1 x_1 + \beta_2 x_2 \)
The residual for each observed \( y \) is:
\( e_i = y_i - \hat{y}_i \)
The sum of squared errors (SSE) and total sum of squares (SST) are computed as in Question 2, considering residuals and deviations, respectively. Estimating \( y \) at specific predictor values and calculating \( R^2 \) involve substituting the values into the derived regression equation.
Question 4: Logistic Regression Odds Ratio Difference
Mr. King’s observed odds ratio is 3/2, and the predicted odds ratio is 2/3. The probabilities corresponding to odds ratios are calculated as:
\( p = \frac{\text{odds}}{1 + \text{odds}} \)
Calculate the observed probability \( p_{obs} \) and predicted probability \( p_{pred} \), then their absolute difference:
\( |p_{obs} - p_{pred}| \)
Question 5: Logistic Regression Framework and Estimation
The logistic regression model relates the natural log of odds to the independent variable:
\( \ln(\text{odds}) = \beta_0 + \beta_1 \times \text{Length} \)
Given the number of positive and negative cases for a given length, the odds ratio can be calculated, then transformed into probability:
\( p = \frac{\text{odds}}{1 + \text{odds}} \)
The relationship between length and probability can be modeled using regression coefficients estimated via the formulas derived from the data, applying logarithmic transformations as needed. To estimate total cases for achieving 100 positive cases at length 5000, use the probability estimate to determine total cases: \( \text{Total Cases} = \frac{100}{p} \).
Conclusion
Comprehensive understanding and manual calculation techniques for covariance, correlation, regression analysis, and logistic regression are vital for interpreting data and making predictions. These techniques are widely applicable across fields such as marketing, manufacturing, healthcare, and technology. Proper application of formulas, careful data handling, and critical analytical thinking enable accurate, meaningful insights from raw data.
References
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